Independent domination and vertex coloring in graphs

Let G = (V,E) be a graph of order n. Let gamma(i)(G),beta(0)(G) and chi(G) denote respectively the independent domination number, independence number and chromatic number of G. These parameters satisfy the well known inequality gamma(i)(G) <= beta(0)(G) <= n - chi(G) + 1. In this paper we determine conditions under which the above inequalities become equalities and use these conditions to determine extremal graphs in some specific graph classes.